Total 11 Posts

Great Classroom Action

It’s summer break here in North America so there’s very little classroom action, much less great classroom action. Let me take advantage of the lull and empty out the links that collected in my link trap last winter.

Bryan Meyer asks what you do when your students get into a math fight:

There was a big controversy about whether or not we should count 1+2 and 2+1 as two different options or if we should just count them together as one option. [..] Neither side was willing to budge, so I suggested we conduct a huge experiment with lots of trials across all three of my classes so that we could put the results together and see what conclusions we could draw.

Tom Ward asks a question that’s strangely compelling, “Should you drive or fly?

So now, instead of simply plotting flight cost vs. distance, weâ€™ll also plot driving cost vs. distance, helping us find the answer to the question above. A couple plots, a couple lines of best fit, an intersection representing a break even point and yes, you can use this in real life.

Scott Farrar wins the Emmy for best graph.

Heather Kohn creates a ruckus in homeroom with the jelly bean guessing contest:

All eyes were on the board as I highlighted the data, clicked the insert tab, and chose the first scatterplot option. There were many â€œWhat?! How?!â€ comments as students digested the graph before them. They immediately wanted to know if the graph would always be in that shape, and this led us into our discussion of graphing absolute value functions and describing their characteristics (over the next two class periods).

[Makeover] Postage Rates

This is from Pearson’s Algebra 1 textbook for iPad.

What I Did

• Establish a need for the graph, in general. Why are we drawing a graph? What’s the point? Does my ability to draw a graph serve any larger purpose than getting me points on an assignment? This task doesn’t have an answer to that question.
• Establish a need for the step graph, in particular. Why are we drawing a step graph? What’s the point. Does the step graph have any advantage over other graphs? This task doesn’t have an answer to that question.

Tom Ward has me covered on the first point. Nothing’s topping this aspirational save-the-date for his 2019 marriage to Ms. Stone. What will postage cost then?

Graphs and equations of data are useful when they let us predict something external to the data we know. We don’t know the price of postage stamps in 2019 so we can extend a linear model beyond the data and find out what it might be.

Mr. Ward will need to scrounge up 54 cents per invitation.

But he still hasn’t given us a reason to care about the step graph. For that we look at internal data. We tell the kid, “Hey, your graph is messed up. If you hand that graph to someone, it says the cost of postage in 2003 was 39 cents and the cost in 2005 was 41 cents. But the cost in both those years was only 37 cents.”

If you’re going to make a graph that tells the story of the data accurately you’re going to need a different model than a straight line. Enter the step.

What You Did

Aside from Tom Ward’s superb work, over on the blogs:

• Scott Hills seizes the opportunity to show students the benefits of a well-scaled axis.
• Beth Ferguson removes step graphs from the task, which is one way to handle the problem.
• Evan Weinberg goes digital, though I’m not sure what the digital medium adds here. He also just asserts that the student’s graph “should be a step function,” which highlights the difficulty again of motivating a need for this function family.

Featured Comment

Instead of a wedding invitation, change it to a graduation invitation. Have the kids estimate how many invitations they would have to mail out. They could then calculate the cost of their invitations. You could also have them calculate the cost that their parents/grandparents paid for their graduation invitations.

Computers Are Not A Natural Medium For Doing Mathematics, Ctd.

You may have heard that San Jose State University’s recent partnership with Udacity ended with MOOC-enrolled students passing courses at much lower rates than their on-campus cohorts. Lots has been said about these results (Phil Hill has a good round-up of the coverage) but there’s one line that deserves more coverage:

When students did get to the online programs, even navigating the computer systems could be daunting. One of the questions that tutors were frequently asked was how to do exponential notation on a computer.

Again we find computers are not a natural medium for doing mathematics. There’s nothing intuitive about pressing Shift + 6 to write an exponent, no inherent connection between the idea and the action. This isn’t true for computer science, where the medium is perfectly suited for the course. Or even for English composition, where typing words is only one intuitive abstraction away from writing them with a pen.

I’d wager 90% of people reading this already know how to type an exponent on a computer. They believe it’s easy enough to teach and I don’t think they’re wrong. But this is only one instance of a problem with a lot of reach. Notation makes math difficult on a computer. But notation also makes math more powerful and interesting. That tension will be very difficult to resolve and, so far, online math providers have generally resolved it in favor of the computer at the expense of math’s interest and power.

In our relentless transition from classroom-based math to computer-based math, these SJSU-Udacity results offer us a chance to pause and ask ourselves, “What’s now missing?”

Previously

Computers Are Not A Natural Medium For Doing Mathematics

2013 Jul 26. Okay, taking friendly fire on Twitter, I posed this challenge:

Use a computer to compose a clear proof that a triangle’s midsegments create similar triangles and send it to me for assessment.

My guess is you’ll find the process a lot less annoying and a lot more clear when you pick up a pencil and some paper.

But on the other hand, the reality is that if our students use math any time later in their life, thereâ€™s a really good likelihood it will involve a computer, whether itâ€™s using Mathematica to solve complex problems, doing computer programming, or just entering formulas in Excel. There is value in learning the notation for entering formulas in a computer, and it provides an valuable side benefit of reinforcing proper syntax to ensure proper order of operation.

Iâ€™ve slowed down on the Euler Project problems, but last I checked, I was just short of 200. Those problems require a computer to do the math, but you donâ€™t do the math on the computer. I have a notebook (ink on paper) dedicated to that that has a couple hundred pages filled with notes and drawings â€“ thatâ€™s the math.

[Makeover] Tire Marks

What I Did

• Reduce extraneous literacy demand. A lot of visual information has been encoded in text. Let’s get that information back in its natural medium.
• Delay the abstraction. Tables and graphs and equations will eventually be useful but let’s delay their introduction until we need them.
• Get a better image. The illustration here is a member of the “job testimonial” genre. ie. “Trooper Bob uses math, so you should too.” I’m unconvinced that message will sway classroom opinion on Algebra even a little. Instead let’s put the student into Trooper Bob’s shoes, doing Trooper Bob’s work.
• Ask a better question. Neither of the two questions here addresses any of Trooper Bob’s concerns. The first has you extend a graph for no discernible purpose. (And why extend the graph from 60 feet to 100 feet. Is that just arbitrary?) The second poses the fantastic scenario where Trooper Bob comes to the scene of a wreck already aware of how fast the car was traveling and then proceeds to do math to figure out the length of the tire marks in front of him. Which he could just measure.

So show this picture of a wreck. Ask your students to guess how fast you think that car was going when it hit the brakes. Tell them they have to figure out if it broke the law. Do they think it was speeding?

Then show them this image.

Ask your students to rank the cars from fastest to slowest. Ask them how they know. They’ve decided the variables “length of skid” and “speed” are positively related. But what kind of relationship is it? This is where a graph â€“ a picture of a relationship â€“ is so useful. Show them the data.

Have them graph the data. This is a little new to us. It isn’t linear. It isn’t quadratic. It isn’t exponential. Offer an explanation of the root model. It’s the inverse of a parabola. With the parabola, a little growth in the horizontal direction results in a lot of growth in the vertical direction. With the root model, a lot of growth is required in the horizontal direction before you get even a little growth in the vertical direction.

Now they can find the exact model for these data and evaluate it for 232.7 feet.

68 miles per hour in a residential zone? You won’t be needing that drivers license for a long time.

What You Did

Over on the blogs:

• Nicholas Chan encourages modeling also, where students make predictions from data.
• Eric Scholz has the same, except where I start with the accident you’re trying to solve and then get smaller data for modeling, he starts by showing students the smaller data and then ending with the accident you’re trying to solve. Is the difference substantial?
• Matthew Jones sends along this clip, which would make for interesting watching after our math work. I’m not sure what work the students would do on the video, though.
• Kate says, “bring the cop to school,” which could be great, but again what math work do the students do?

Featured Comment

Now this is the reason I follow this blog. I am a criminal justice instructor and many students come to my field as fugitives from math and science. Use of these materials is helping me develop criminal justice contextualized math resources for a class proposal.

What We Can Learn About Learning From Khan Academyâ€™s Source Code, Ctd.

I’m used to seeing pedagogy manifest itself in lesson plans and classroom observations and curriculum and videos. It’s interesting, now, to see pedagogical decisions manifest themselves in web design and code also. For example, here’s some Javascript from Khan Academy’s box-and-whisker plot exercises.

Head over to the exercise. Complete a couple. What pedagogical mistake has Khan Academy made in the highlighted lines? How would you fix it?

Don’t get put off by the code. If you’ve taught box-and-whisker plots, you can sort out the issue here.

[via Travis Olson]

Brian lands it:

This code will always generate 15 data points, and these points will not have any outliers (outside 1.5 * (Q3 â€“ Q1)), so students can just pattern match and drag the lines to the 1st, 4th, 8th, 12th, and 15th places once theyâ€™ve sorted the data. Itâ€™s kind of fun the first time.

Dan Anderson piles on:

Agree with Brian. Always 15 data points? Never have to deal with â€œhaving two mediansâ€? Ever? The data is between 0 and 15 (never -40 to -30, never 100 to 1000, never 0.80 to 1.15)? No outliers? Always starting with the data and making a box-and-whisker, never using the box-and-whisker to make conclusions?

Peter Franza picks on a different issue:

I think the largest error is the reliance on random numbers to provide a set of assessments that test an actual set of knowledge.

Random number generators are great for creating a large set of problems that are all basically the same, but in my experience you can provide better assessments/examples with a much smaller set of questions that are designed to illustrate the concept.

Others have danced around it, but the fundamental flaw (as in some, but not all, Khan exercises) is that you get

THE SAME QUESTION

seven straight times, without any change in structure or difficulty, even though the underlying task has a huge variation in structure and difficulty.

Ben Alpert responds from Khan Academy:

Iâ€™ve updated the exercise so that it now includes anywhere from 8 to 15 points, so students are forced to deal with two middle numbers, both in finding the median and in finding the quartiles.